Noether-Symmetry Analysis using Alternative Lagrangian Representations

We have sought to work with an approach to Noether symmetry analysis which uses the properties of infinitesimal point transformations in the space-time (q, t) variable to establish the association between symmetries and conservation laws of a dynamical system. In this approach symmetries are expressed in the form of generators. We have studied the variational or Noether symmetries of two uncoupled Harmonic oscillators and two such oscillators coupled by an interaction. Both these systems can have alternative Lagrangian representations. We have studied in detail how the association between symmetries and conservation laws changes as one alters the analytic or Lagrangian representation. This analysis is carried out with a view to explicitly demonstrate that the correlation between symmetry transformation and corresponding invariant quantity depends crucially on the choice of the analytic representation. PACS 45.20.Jj, 45.20.df, 45.20.dh     

Determination of the Electromagnetic Lagrangian from a System of Poisson Brackets

The Lagrangian and Hamiltonian formulations of electromagnetism are reviewed and the Maxwell equations are obtained from the Hamiltonian for a system of many electric charges. It is shown that three of the equations which were obtained from the Hamiltonian, namely the Lorentz force law and two Maxwell equations, can be obtained as well from a set of postulated Poisson brackets. It is shown how the results derived from these brackets can be used to reconstruct the original Lagrangian for the theory aided by some reasoning based on physical concepts.     

Singular Lagrangian for the polaron

A Lagrangian which describes the electron-phonon interaction is proposed, starting from a simple model of interacting string and charged particle. The Lagrangian is singular, containing both bosonic variables (i.e., phonons) and fermionic ones (i.e., the electron). Symmetry Dirac brackets are used to obtain the BCS Hamiltonian. The addition of a total derivative of a scalar function to the Lagrangian density doesnot alter the quantization procedure.     

Lagrangian instanton for the Kraichnan model

We consider high-order correlation functions of the passive scalar in the Kraichnan model. Using the instanton formalism, we find the scaling exponents ζ_n of the structure functions S _n for n≫1 under the additional condition dζ₂≫1 (where d is the dimensionality of space).At n<n _c (where n _c =dζ₂/[2(2−ζ₂)]) the exponents are ζ_n=(ζ ₂/4)(2n−n ²/n _c ), while at n>n _c they are n-independent: ζ _n=ζ₂ n _c /4. We also estimate the n-dependent factors in S _n . Pis’ma Zh. éksp. Teor. Fiz. 68, No. 7, 588–593 (10 October 1998) Published in English in the original Russian journal. Edited by Steve Torstveit.     

电磁波作用下分子极化率的量子力学表示

由Rayleigh和Raman散射的跃迁几率幅,引入电磁波作用下分子极化率的量子力学表示。表明极化率所感生的偶极子如何成为Rayleigh和Raman散射的源并证明在Born-Oppenheimer近似下,极化率只与分子的电子运动有关。     

Exact propagator for the one-dimensional time-dependent quadratic Lagrangian

Extending Montroll's method [Montroll, E. W., Commun. Pure Appl. Math. 5, 415 (1952)] to the one-dimensional time-dependent quadratic Lagrangian, we first evaluate the propagator beyond caustics by including Maslov's correction factor. The propagator at caustics is then obtained by introducing the modified semigroup property. Finally, the wave functions are derived from the propagator obtained.     

Lagrangian Markovianized field approximation for turbulence

In a previous communication (W.J.T. Bos and J.-P. Bertoglio 2006, Phys. Fluids, 18, 031706), a self-consistent Markovian triadic closure was presented. The detailed derivation of this closure is given here, relating it to the Direct Interaction Approximation and Quasi-Normal types of closure. The time-scale needed to obtain a self-consistent closure for both the energy spectrum and the scalar variance spectrum is determined by evaluating the correlation between the velocity and an advected displacement vector-field. The relation between this latter correlation and the velocity–scalar correlation is stressed, suggesting a simplified model of the latter. The resulting closed equations are numerically integrated and results for the energy spectrum, scalar fluctuation spectrum and velocity–displacement correlation spectrum are presented for low, unity and high values of the Schmidt number.     

Effective low energy Lagrangian for QCD

The peculiar collective excitations of gluons, related to the Lorentz transformations of the conjugate field tensor, were proposed in a previous paper. The pseudoscalar mesons are related in the same way to the chiral transformations of quarks. We discuss these relations, including the U (1) problem, and derive the low energy effective Lagrangian for all these collective excitations. This Lagrangian contains the potential for the U (1) field due to axial anomalies.     

Euler-Heisenberg-Schwinger Lagrangian for nonadiabatically varying fields

Nonperturbative particle production in external variable fields is important in astrophysics. Although a wide range of techniques exists for calculating production rate, none of them can handle exactly the case of a strongly inhomogeneous field. Here, an electric field with an abrupt switch-on is considered. First, a standard semiclassical technique is used. Then, a scattering problem approach to this case is developed and time-dependent corrections to the effective Euler-Heisenberg-Schwinger Lagrangian are calculated. The text was submitted by the author in English.     

Equivariant Asymptotics for Bohr-Sommerfeld Lagrangian Submanifolds

Suppose given a complex projective manifold M with a fixed Hodge form Ω. The Bohr-Sommerfeld Lagrangian submanifolds of (M,Ω) are the geometric counterpart to semi-classical physical states, and their geometric quantization has been extensively studied. Here we revisit this theory in the equivariant context, in the presence of a compatible (Hamiltonian) action of a connected compact Lie group.     

On the Gauge Invariant Lagrangian for Seiberg--Witten Topological Monopoles

A topological gauge invariant Lagrangian for Seiberg--Witten monopoleequations is constructed. The action is invariant under a huge classof gauge transformations which after BRST fixing leads to the BRSTinvariant action associated to Seiberg--Witten monopole topologicaltheory. The supersymmetric transformation of the fields involved inthe construction is obtained from the nilpotent BRST algebra.     

Relativistic corrections to the Lagrangian for interacting charged particles

We give the Lagrangian of a system of moving charged particles up to the fourth approximation in 1/c neglecting dipole radiation of the system. In this case the appearance of the electromagnetic waves (quadrupole radiation) by moving charges occurs in the fifth approximation in 1/c.     

Dirac-Hestenes Lagrangian

We formulate the variational principle of theDirac equation within the noncommutative even space-timesubalgebra, the Clifford -algebra . A fundamental ingredient in ourmultivectorial algebraic formulation is a -complex geometry, . We derive the Lagrangian for theDirac-Hestenes equation and show that it must be mapped on , where denotes an -algebra of functions.     

Generalization of the Darwin Lagrangian

Starting from a Lagrangian treatment of classical electrodynamics and using simple heuristic arguments, an effective generalization of the Darwin Lagrangian for point charges moving with arbitrary velocities (v<c) is obtained. In addition, another, the simplesta priori Lagrangian (preserving the most important features of standard classical theory) is proposed.     

A quadratic spinor Lagrangian for general relativity

We present a newfinite action for Einstein gravity in which the Lagrangian is quadratic in the covariant derivative of a spinor field. Via a new spinor-curvature identity, it is related to the standard Einstein-Hilbert Lagrangian by a total differential term. The corresponding Hamiltonian, like the one associated with the Witten positive energy proof, is fully 4-covariant. It defines quasi-local energy-momentum and can be reduced to the one in our recent positive energy proof.This essay received the fourth award from the Gravity Research Foundation, 1994—Ed.